In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry.
In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield (composed of a complex scalar and a spinor fermion) whose cubic superpotential leads to a renormalizable theory.
The treatment in this article largely follows that of Figueroa-O'Farrill's lectures on supersymmetry,[2] and to some extent of Tong.
It is arguably the simplest supersymmetric field theory in four dimensions, and is ungauged.
In a preliminary treatment, the theory is defined on flat spacetime (Minkowski space).
The matter content is a real scalar field
This is a preliminary treatment in the sense that the theory is written in terms of familiar scalar and spinor fields which are functions of spacetime, without developing a theory of superspace or superfields, which appear later in the article.
The Lagrangian of the free, massless Wess–Zumino model is where The corresponding action is Supersymmetry is preserved when adding a mass term of the form Supersymmetry is preserved when adding an interaction term with coupling constant
are combined into a single complex scalar field
Defining the superpotential the Wess–Zumino action can also be written (possibly after relabelling some constant factors)
, one finds that this is a theory with a massive complex scalar
, which are all familiar interactions from courses in non-supersymmetric quantum field theory.
The 'spin coordinates' are so called not due to any relation to angular momentum, but because they are treated as anti-commuting numbers, a property typical of spinors in quantum field theory due to the spin statistics theorem.
Defining the supercovariant derivative a chiral superfield satisfies
The field content is then simply a single chiral superfield.
However, the chiral superfield contains fields, in the sense that it admits the expansion with
This allows recovery of the preliminary forms, after eliminating the non-dynamical
, the action (for the free, massless Wess–Zumino model) takes on the simple form where
is complex, to ensure the action is real its conjugate must also be added.
The action is invariant under the supersymmetry transformations, given in infinitesimal form by where
The alternative form is invariant under the transformation Without developing a theory of superspace transformations, these symmetries appear ad-hoc.
The massless Wess–Zumino model admits a larger set of symmetries, described at the algebra level by the superconformal algebra.
The action generalizes straightforwardly to multiple chiral superfields
The most general renormalizable theory is where the superpotential is where implicit summation is used.
There is residual freedom to make a unitary transformation in order to diagonalise the mass matrix
, if the multiplet is massive then the Weyl fermion has a Majorana mass.
the two Weyl fermions can have a Dirac mass, when the superpotential is taken to be
This symmetry can be gauged and coupled to supersymmetric Yang–Mills to form a supersymmetric analogue to quantum chromodynamics, known as super QCD.
using the equations of motion, the following expression is obtained: where and A superpotential
can be added to form the more general action where the Hessians of